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Simplificación de expresiones/ Descomposición en fracciones simples/ (tan(p/(4+a))+tan(a-p/4))/tan(2*a)

¿Cómo vas a descomponer esta (tan(p/(4+a))+tan(a-p/4))/tan(2*a) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
   /  p  \      /    p\
tan|-----| + tan|a - -|
   \4 + a/      \    4/
-----------------------
        tan(2*a)
tan⁡(pa+4)+tan⁡(a−p4)tan⁡(2a)\frac{\tan{\left(\frac{p}{a + 4} \right)} + \tan{\left(a - \frac{p}{4} \right)}}{\tan{\left(2 a \right)}} 
(tan(p/(4 + a)) + tan(a - p/4))/tan(2*a)
Unión de expresiones racionales [src] 
   /-p + 4*a\      /  p  \
tan|--------| + tan|-----|
   \   4    /      \4 + a/
--------------------------
         tan(2*a)
tan⁡(pa+4)+tan⁡(4a−p4)tan⁡(2a)\frac{\tan{\left(\frac{p}{a + 4} \right)} + \tan{\left(\frac{4 a - p}{4} \right)}}{\tan{\left(2 a \right)}} 
(tan((-p + 4*a)/4) + tan(p/(4 + a)))/tan(2*a)
Potencias [src] 
   /  /     /    p\      /     p\\     /    I*p     -I*p \\                    
   |  |   I*|a - -|    I*|-a + -||     |   -----    -----||                    
   |  |     \    4/      \     4/|     |   4 + a    4 + a||                    
   |I*\- e          + e          /   I*\- e      + e     /| / -2*I*a    2*I*a\ 
-I*|------------------------------ + ---------------------|*\e       + e     / 
   |      /    p\      /     p\           I*p     -I*p    |                    
   |    I*|a - -|    I*|-a + -|          -----    -----   |                    
   |      \    4/      \     4/          4 + a    4 + a   |                    
   \   e          + e                   e      + e        /                    
-------------------------------------------------------------------------------
                                  2*I*a    -2*I*a                              
                               - e      + e
−i(i(ei(−a+p4)−ei(a−p4))ei(−a+p4)+ei(a−p4)+i(−eipa+4+e−ipa+4)eipa+4+e−ipa+4)(e2ia+e−2ia)−e2ia+e−2ia- \frac{i \left(\frac{i \left(e^{i \left(- a + \frac{p}{4}\right)} - e^{i \left(a - \frac{p}{4}\right)}\right)}{e^{i \left(- a + \frac{p}{4}\right)} + e^{i \left(a - \frac{p}{4}\right)}} + \frac{i \left(- e^{\frac{i p}{a + 4}} + e^{- \frac{i p}{a + 4}}\right)}{e^{\frac{i p}{a + 4}} + e^{- \frac{i p}{a + 4}}}\right) \left(e^{2 i a} + e^{- 2 i a}\right)}{- e^{2 i a} + e^{- 2 i a}} 
-i*(i*(-exp(i*(a - p/4)) + exp(i*(-a + p/4)))/(exp(i*(a - p/4)) + exp(i*(-a + p/4))) + i*(-exp(i*p/(4 + a)) + exp(-i*p/(4 + a)))/(exp(i*p/(4 + a)) + exp(-i*p/(4 + a))))*(exp(-2*i*a) + exp(2*i*a))/(-exp(2*i*a) + exp(-2*i*a))
Abrimos la expresión [src] 
                                 /  p  \                                               /p\                       /  p  \             2       /p\      
                              tan|-----|                3                           tan|-|             tan(a)*tan|-----|          tan (a)*tan|-|      
           tan(a)                \4 + a/             tan (a)                           \4/                       \4 + a/                     \4/      
--------------------------- + ---------- - --------------------------- - --------------------------- - ----------------- + ---------------------------
                2       /p\    2*tan(a)                    2       /p\                   2       /p\           2                           2       /p\
2*tan(a) + 2*tan (a)*tan|-|                2*tan(a) + 2*tan (a)*tan|-|   2*tan(a) + 2*tan (a)*tan|-|                       2*tan(a) + 2*tan (a)*tan|-|
                        \4/                                        \4/                           \4/                                               \4/
−tan⁡(a)tan⁡(pa+4)2+tan⁡(pa+4)2tan⁡(a)−tan⁡3(a)2tan⁡2(a)tan⁡(p4)+2tan⁡(a)+tan⁡2(a)tan⁡(p4)2tan⁡2(a)tan⁡(p4)+2tan⁡(a)+tan⁡(a)2tan⁡2(a)tan⁡(p4)+2tan⁡(a)−tan⁡(p4)2tan⁡2(a)tan⁡(p4)+2tan⁡(a)- \frac{\tan{\left(a \right)} \tan{\left(\frac{p}{a + 4} \right)}}{2} + \frac{\tan{\left(\frac{p}{a + 4} \right)}}{2 \tan{\left(a \right)}} - \frac{\tan^{3}{\left(a \right)}}{2 \tan^{2}{\left(a \right)} \tan{\left(\frac{p}{4} \right)} + 2 \tan{\left(a \right)}} + \frac{\tan^{2}{\left(a \right)} \tan{\left(\frac{p}{4} \right)}}{2 \tan^{2}{\left(a \right)} \tan{\left(\frac{p}{4} \right)} + 2 \tan{\left(a \right)}} + \frac{\tan{\left(a \right)}}{2 \tan^{2}{\left(a \right)} \tan{\left(\frac{p}{4} \right)} + 2 \tan{\left(a \right)}} - \frac{\tan{\left(\frac{p}{4} \right)}}{2 \tan^{2}{\left(a \right)} \tan{\left(\frac{p}{4} \right)} + 2 \tan{\left(a \right)}} 
tan(a)/(2*tan(a) + 2*tan(a)^2*tan(p/4)) + tan(p/(4 + a))/(2*tan(a)) - tan(a)^3/(2*tan(a) + 2*tan(a)^2*tan(p/4)) - tan(p/4)/(2*tan(a) + 2*tan(a)^2*tan(p/4)) - tan(a)*tan(p/(4 + a))/2 + tan(a)^2*tan(p/4)/(2*tan(a) + 2*tan(a)^2*tan(p/4))
Respuesta numérica [src] 
(tan(p/(4 + a)) + tan(a - p/4))/tan(2*a)
(tan(p/(4 + a)) + tan(a - p/4))/tan(2*a)
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